Measures of Central Tendency

Researchers are often interested in defining a value that best describes some attribute of the population. Often this attribute is a measure of central tendency or a proportion.

Measures of Central Tendency

Several different measures of central tendency are defined below.

  • The mode is the most frequently appearing value in the population or sample. Suppose we draw a sample of five women and measure their weights. They weigh 100 pounds, 100 pounds, 130 pounds, 140 pounds, and 150 pounds. Since more women weigh 100 pounds than any other weight, the mode would equal 100 pounds.

  • To find the median, we arrange the observations in order from smallest to largest value. If there is an odd number of observations, the median is the middle value. If there is an even number of observations, the median is the average of the two middle values. Thus, in the sample of five women, the median value would be 130 pounds; since 130 pounds is the middle weight.

  • The mean of a sample or a population is computed by adding all of the observations and dividing by the number of observations. Returning to the example of the five women, the mean weight would equal (100 + 100 + 130 + 140 + 150)/5 = 620/5 = 124 pounds.

Proportions and Percentages

When the focus is on the degree to which a population possesses a particular attribute, the measure of interest is a percentage or a proportion.

  • A proportion refers to the fraction of the total that possesses a certain attribute. For example, we might ask what proportion of women in our sample weigh less than 135 pounds. Since 3 women weigh less than 135 pounds, the proportion would be 3/5 or 0.60.

  • A percentage is another way of expressing a proportion. A percentage is equal to the proportion times 100. In our example of the five women, the percent of the total who weigh less than 135 pounds would be 100 * (3/5) or 60 percent.

Notation

Of the various measures, the mean and the proportion are most important. The notation used to describe these measures appears below:

  • X: Refers to a population mean.
  • x: Refers to a sample mean.
  • P: The proportion of elements in the population that has a particular attribute.
  • p: The proportion of elements in the sample that has a particular attribute.
  • Q: The proportion of elements in the population that does not have a specified attribute. Note that Q = 1 - P.
  • q: The proportion of elements in the sample that does not have a specified attribute. Note that q = 1 - p.

Note that capital letters refer to population parameters, and lower-case letters refer to sample statistics.